In this shear mapping of an image of the Mona Lisa, the picture was deformed in such a way that its central vertical axis was not modified.
In mathematics, a shear or transvection is a particular kind of linear mapping. Its effect leaves fixed all points on one axis and other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from the axis. It is notable that shear mappings carry areas into equal areas.
Elementary form
In the plane {(x,y): x,y - R }, a vertical shear (or shear parallel to the x axis) for m - 0 of vertical lines x = a into lines y = (x - a)/m of slope 1/m is represented by the linear mapping
One can substitute 1/m for m in the matrix to get lines y = m(x - a) of slope m if desired.
A horizontal shear (or shear parallel to the y axis) of lines y = b into lines y = mx + b is accomplished by the linear mapping
These are special cases of shear matrices, which allow for generalization to higher dimensions. The shear elements here are either m or 1/m, case depending.
Advanced form
For a vector space V and subspace W, a shear fixing W translates all vectors parallel to W.
To be more precise, if V is the direct sum of W and W-, and we write vectors as
- v = w + w-
correspondingly, the typical shear fixing W is L where
- L(v) = (w + w-M) + w-
where M is a linear mapping from W- into W. Therefore in block matrix terms L can be represented as
with blocks on the diagonal I (identity matrix), with M below the diagonal, and 0 above.
See also
References
- Weisstein, Eric W. "Shear" from Mathworld, A Wolfram Web Resource.
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